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A note on Steinerberger's curvature for graphs

David Cushing, Supanat Kamtue, Erin Law, Shiping Liu, Florentin Münch, Norbert Peyerimhoff

TL;DR

This note develops explicit Steinerberger curvature formulas for block graphs, investigates how curvature behaves under bridging graphs by a single edge, and characterizes Bonnet-Myers sharp graphs in terms of antipodality and self-centeredness. It derives exact curvature expressions for block graphs, shows that joining graphs by a bridge enforces proportional curvature distributions with computable constants, and provides a direct Bonnet-Myers bound along with a complete equivalence between antipodal graphs and self-centered Bonnet-Myers sharp graphs. The work also situates Steinerberger curvature relative to Ollivier–Ricci curvature, discusses nonpositive curvature phenomena via experimental examples, and highlights questions about curvature dynamics under leaf-attachments. Overall, it advances explicit, computable curvature tools for finite graphs and clarifies how global curvature interacts with graph composition and symmetry.

Abstract

In this note, we provide Steinerberger curvature formulas for block graphs, discuss curvature relations between two graphs and the graph obtained by connecting them via a bridge, and show that self-centered Bonnet-Myers sharp graphs are precisely those which are antipodal. We also discuss similarities and differences between Steinerberger and Ollivier Ricci curvature results.

A note on Steinerberger's curvature for graphs

TL;DR

This note develops explicit Steinerberger curvature formulas for block graphs, investigates how curvature behaves under bridging graphs by a single edge, and characterizes Bonnet-Myers sharp graphs in terms of antipodality and self-centeredness. It derives exact curvature expressions for block graphs, shows that joining graphs by a bridge enforces proportional curvature distributions with computable constants, and provides a direct Bonnet-Myers bound along with a complete equivalence between antipodal graphs and self-centered Bonnet-Myers sharp graphs. The work also situates Steinerberger curvature relative to Ollivier–Ricci curvature, discusses nonpositive curvature phenomena via experimental examples, and highlights questions about curvature dynamics under leaf-attachments. Overall, it advances explicit, computable curvature tools for finite graphs and clarifies how global curvature interacts with graph composition and symmetry.

Abstract

In this note, we provide Steinerberger curvature formulas for block graphs, discuss curvature relations between two graphs and the graph obtained by connecting them via a bridge, and show that self-centered Bonnet-Myers sharp graphs are precisely those which are antipodal. We also discuss similarities and differences between Steinerberger and Ollivier Ricci curvature results.
Paper Structure (5 sections, 8 theorems, 76 equations, 6 figures)

This paper contains 5 sections, 8 theorems, 76 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Block graphs
  3. Bridges
  4. Questions related to non-positive Steinerberger curvature
  5. Bonnet-Myers sharp graphs

Key Result

Theorem 1

Let $G=(V,E)$ be a block graph consisting of $r$ blocks, each of size $(P_i)_{i=1}^r$ and Let $x \in V$ be contained in $s$ blocks labeled by $i_1,\dots,i_s$. Let Then we have

Figures (6)

  • Figure 1:
  • Figure 2: The graph family $A(n)$: $A(1), A(2), A(3), A(4), A(5)$ from left to right
  • Figure 3: A graph $G$ with 9 vertices and vanishing total Steinerberger curvature (left) and an extension of $G$ by consecutive attachment of $8$ leaves (right)
  • Figure 4: A path length $5$ (left) needs $7$ leaves, a cycle of length $6$ (center) needs $6$ leaves, and the graph with $5$ vertices (right) needs $4$ leaves to make all their vertices negatively curved.
  • Figure 5: The distance-balanced and Bonnet-Myers sharp Handa graph with vertex enumeration (left) and its Steinerberger curvatures (right)
  • ...and 1 more figures

Theorems & Definitions (26)

  • Theorem 1
  • proof
  • Example 2
  • Remark 3
  • Theorem 4
  • proof
  • Remark 5
  • Theorem 6
  • Remark 7
  • proof : Proof of Theorem \ref{['thm:BB_ext2']}
  • ...and 16 more